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Localization of Closed (or Periodic) Solutions of a Differential System with Concave Nonlinearities
Author(s) -
Sandqvist Allan,
Andersen Kurt Munk
Publication year - 2005
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s002460930400390x
Subject(s) - mathematics , concave function , convexity , regular polygon , combinatorics , mathematical analysis , convex function , scalar (mathematics) , exponent , pure mathematics , differential equation , hölder condition , discrete mathematics , geometry , linguistics , philosophy , financial economics , economics
Consider a scalar differential equationx ˙ = f ( t , x ) , ( t , x ) ∈ I × R , where I is an open interval containing [0, T ]. Assume that f ( t , x ) is continuous with a continuous derivativef x ′ ( t , x ) , and weakly concave (or weakly convex) in x for all t ∈ I , though strictly concave (or strictly convex) for some t ∈ [0, T ]. It is well known that in this case there can be either no, one or two closed solutions; that is, solutions ϕ( t ) for which ϕ(0) = ϕ( T ) If there are two closed solutions, then the greater has a negative characteristic exponent and the smaller has a positive one. It is easily seen that this is equivalent to a statement on localization of closed solutions. It is shown how this statement can be generalized to systems of differential equationsx ˙ _ = f _ ( t , x _ ) , ( t , x _ ) ∈ I × R n . The requirements are that the coordinate functionsf j ( t , x _ ) ) be continuous with continuous derivatives with respect to x 1 , x 2 , …, x n , that the f j are weakly concave (or weakly convex) in x _ , and that a certain condition pertaining to strict concavity (or strict convexity) is fulfilled. 2000 Mathematics Subject Classification 34C25, 34C12.